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September  2014, 34(9): 3847-3873. doi: 10.3934/dcds.2014.34.3847

## Dimension estimates in non-conformal setting

 1 Department of mathematics, Soochow University, Suzhou 215006, Jiangsu, China 2 Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, China

Received  July 2013 Revised  December 2013 Published  March 2014

Given an arbitrary subset of a non-conformal repeller of a $C^1$ map. Using directly the definitions of Hausdorff dimension and Box dimension and of pressure, this paper first proves that the zeros of the topological pressure on this set give its dimension estimates. And without using the estimate of pointwise dimension of an ergodic measure on a non-conformal repeller, it is showed that the zeros of non-additive measure-theoretic pressure give the lower and upper bound of dimension estimate for it. Some results from [22,41] are extended for $C^1$ maps or arbitrary subsets of a non-conformal repeller. And a remark on Rugh's result [34] is also given in this paper.
Citation: Juan Wang, Yongluo Cao, Yun Zhao. Dimension estimates in non-conformal setting. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3847-3873. doi: 10.3934/dcds.2014.34.3847
##### References:
 [1] J. Ban, Y. Cao and H. Hu, The dimensions of a non-conformal repeller and an average conformal repeller,, Trans. Amer. Math. Soc., 362 (2010), 727.  doi: 10.1090/S0002-9947-09-04922-8.  Google Scholar [2] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems,, Ergod. Th. Dynam. Syst., 16 (1996), 871.  doi: 10.1017/S0143385700010117.  Google Scholar [3] L. Barreira, Dimension estimates in nonconformal hyperbolic dynamics,, Nonlinearity, 16 (2003), 1657.  doi: 10.1088/0951-7715/16/5/307.  Google Scholar [4] L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics,, Progress in Mathematics, (2008).   Google Scholar [5] L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory,, Progress in Mathematics, (2011).  doi: 10.1007/978-3-0348-0206-2.  Google Scholar [6] L. Barreira and K. Gelfert, Dimension estimates in smooth dynamics: A survey of recent results,, Ergod. Th. Dynam. Syst., 31 (2011), 641.  doi: 10.1017/S014338571000012X.  Google Scholar [7] L. Barreira and C. Wolf, Pointwise dimension and ergodic decompositions,, Ergod. Th. Dynam. Syst., 26 (2006), 653.  doi: 10.1017/S0143385705000672.  Google Scholar [8] L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension,, Israel J. Math., 116 (2000), 29.  doi: 10.1007/BF02773211.  Google Scholar [9] R. Bowen, Hausdorff dimension of quasicircles,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11.   Google Scholar [10] R. Bowen, Equlibrium States and the Ergodic Theory of Anosov Diffeomorphism,, Lecture Notes in Mathematics, (1975).   Google Scholar [11] M. Brin and A. Katok, On Local Entropy,, in Geometric dynamics (Rio de Janeiro, 1007 (1983), 30.  doi: 10.1007/BFb0061408.  Google Scholar [12] Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials,, Discret. Contin. Dynam. Syst., 20 (2008), 639.   Google Scholar [13] Y. Cao, H. Hu and Y. Zhao, Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure,, Ergod. Th. Dynam. Syst., 33 (2013), 831.  doi: 10.1017/S0143385712000090.  Google Scholar [14] Y. Cao, Dimension spectrum of asymptotically additive potentials for $C^1$ average conformal repellers,, Nonlinearity, 26 (2013), 2441.  doi: 10.1088/0951-7715/26/9/2441.  Google Scholar [15] J. Chen and Y. Pesin, Dimension of non-conformal repellers: A survey,, Nonlinearity, 23 (2010).  doi: 10.1088/0951-7715/23/4/R01.  Google Scholar [16] W. Cheng, Y. Zhao and Y. Cao, Pressures for asymptotically subadditive potentials under a mistake funciton,, Discret. Contin. Dyn. Syst., 32 (2012), 487.  doi: 10.3934/dcds.2012.32.487.  Google Scholar [17] V. Climenhaga, Bowen's equation in the non-uniform setting,, Ergod. Th. Dynam. Syst., 31 (2011), 1163.  doi: 10.1017/S0143385710000362.  Google Scholar [18] M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps,, Nonlinearity, 4 (1991), 103.  doi: 10.1088/0951-7715/4/1/008.  Google Scholar [19] K. Falconer, Fractal Geometry-Mathematical Foundations and Applications,, Second edition. John Wiley & Sons, (2003).  doi: 10.1002/0470013850.  Google Scholar [20] K. Falconer, Bounded distortion and dimension for non-conformal repellers,, Math. Proc. Camb. Phil. Soc., 115 (1994), 315.  doi: 10.1017/S030500410007211X.  Google Scholar [21] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps,, Ergod. Th. Dynam. Syst., 17 (1997), 147.  doi: 10.1017/S0143385797060987.  Google Scholar [22] H. Hu, Dimensions of invariant sets of expanding maps,, Commun. Math. Phys., 176 (1996), 307.  doi: 10.1007/BF02099551.  Google Scholar [23] W. Huang and P. Zhang, Pointwise dimension, entropy and Lyapunov exponents for $C^1$ map,, Trans. Amer. Math. Soc., 364 (2012), 6355.  doi: 10.1090/S0002-9947-2012-05527-9.  Google Scholar [24] F. Ledrappier, Some relations between dimension and Lyapunov exponent,, Commun. Math. Phys., 81 (1981), 229.  doi: 10.1007/BF01208896.  Google Scholar [25] V. Mayer and M. Urbański, Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order,, Ergod. Th. Dynam. Syst., 28 (2008), 915.  doi: 10.1017/S0143385707000648.  Google Scholar [26] V. Mayer and M. Urbański, Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order,, Mem. Amer. Math. Soc., 203 (2010).  doi: 10.1090/S0065-9266-09-00577-8.  Google Scholar [27] V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Lyapnov exponents of dynamical systems,, Trudy Moskov. Mat. Obšž, 19 (1968), 179.   Google Scholar [28] Y. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications,, Chicago: University of Chicago Press, (1997).   Google Scholar [29] F. Przytycki, J. Letelier and S. Smirnov, Equivalence and topo- logical invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps,, Invent. Math., 151 (2003), 29.  doi: 10.1007/s00222-002-0243-x.  Google Scholar [30] F. Przytycki, J. Letelier and S. Smirnov, Equality of pressures for rational functions,, Ergod. Th. Dynam. Syst., 24 (2004), 891.  doi: 10.1017/S0143385703000385.  Google Scholar [31] F. Przytycki and M. Urbažski, Conformal Fractals: Ergodic Theory Methods,, London Mathematical Society, (2010).   Google Scholar [32] D. Ruelle, Repellers for real analytic maps,, Ergod. Th. Dynam. Syst., 2 (1982), 99.  doi: 10.1017/S0143385700009603.  Google Scholar [33] D. Ruelle, An inequality for the entropy of differential maps,, Bol. Soc. Bras. De Mat., 9 (1978), 83.  doi: 10.1007/BF02584795.  Google Scholar [34] H. Rugh, On the dimensions of conformal repellers. Randomness and parameter dependency,, Annals of Mathematics, 168 (2008), 695.  doi: 10.4007/annals.2008.168.695.  Google Scholar [35] M. Urbažski, On the Hausdorff dimension of a Julia set with a rationally indifferent periodic point,, Studia Math., 97 (1991), 167.   Google Scholar [36] M. Urbažski, Parabolic Cantor sets,, Fund. Math., 151 (1996), 241.   Google Scholar [37] M. Urbažski and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family,, Ergod. Th. Dynam. Syst., 24 (2004), 279.  doi: 10.1017/S0143385703000208.  Google Scholar [38] P. Walters, An Introduction to Ergodic Theory,, Springer-Verlag, (1982).   Google Scholar [39] C. Wolf, On the box dimension of an invariant set,, Nonlinearity, 14 (2001), 73.  doi: 10.1088/0951-7715/14/1/303.  Google Scholar [40] L. S. Young, Dimension, entropy and Lyapunov exponents,, Ergod. th. Dynam. Syst., 2 (1982), 109.  doi: 10.1017/S0143385700009615.  Google Scholar [41] Y. Zhang, Dynamical upper bounds for Hausdorff dimension of invariant sets,, Ergod. Th. Dynam. Syst., 17 (1997), 739.  doi: 10.1017/S0143385797085003.  Google Scholar [42] Y. Zhao and Y. Cao, Measure-theoretic pressure for subadditive potentials,, Nonlinear analysis, 70 (2009), 2237.  doi: 10.1016/j.na.2008.03.003.  Google Scholar [43] Y. Zhao, A note on the measure-theoretic pressure in subadditive case,, Chinese Annals of Math., 29 (2008), 325.   Google Scholar

show all references

##### References:
 [1] J. Ban, Y. Cao and H. Hu, The dimensions of a non-conformal repeller and an average conformal repeller,, Trans. Amer. Math. Soc., 362 (2010), 727.  doi: 10.1090/S0002-9947-09-04922-8.  Google Scholar [2] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems,, Ergod. Th. Dynam. Syst., 16 (1996), 871.  doi: 10.1017/S0143385700010117.  Google Scholar [3] L. Barreira, Dimension estimates in nonconformal hyperbolic dynamics,, Nonlinearity, 16 (2003), 1657.  doi: 10.1088/0951-7715/16/5/307.  Google Scholar [4] L. Barreira, Dimension and Recurrence in Hyperbolic Dynamics,, Progress in Mathematics, (2008).   Google Scholar [5] L. Barreira, Thermodynamic Formalism and Applications to Dimension Theory,, Progress in Mathematics, (2011).  doi: 10.1007/978-3-0348-0206-2.  Google Scholar [6] L. Barreira and K. Gelfert, Dimension estimates in smooth dynamics: A survey of recent results,, Ergod. Th. Dynam. Syst., 31 (2011), 641.  doi: 10.1017/S014338571000012X.  Google Scholar [7] L. Barreira and C. Wolf, Pointwise dimension and ergodic decompositions,, Ergod. Th. Dynam. Syst., 26 (2006), 653.  doi: 10.1017/S0143385705000672.  Google Scholar [8] L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension,, Israel J. Math., 116 (2000), 29.  doi: 10.1007/BF02773211.  Google Scholar [9] R. Bowen, Hausdorff dimension of quasicircles,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11.   Google Scholar [10] R. Bowen, Equlibrium States and the Ergodic Theory of Anosov Diffeomorphism,, Lecture Notes in Mathematics, (1975).   Google Scholar [11] M. Brin and A. Katok, On Local Entropy,, in Geometric dynamics (Rio de Janeiro, 1007 (1983), 30.  doi: 10.1007/BFb0061408.  Google Scholar [12] Y. Cao, D. Feng and W. Huang, The thermodynamic formalism for sub-additive potentials,, Discret. Contin. Dynam. Syst., 20 (2008), 639.   Google Scholar [13] Y. Cao, H. Hu and Y. Zhao, Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure,, Ergod. Th. Dynam. Syst., 33 (2013), 831.  doi: 10.1017/S0143385712000090.  Google Scholar [14] Y. Cao, Dimension spectrum of asymptotically additive potentials for $C^1$ average conformal repellers,, Nonlinearity, 26 (2013), 2441.  doi: 10.1088/0951-7715/26/9/2441.  Google Scholar [15] J. Chen and Y. Pesin, Dimension of non-conformal repellers: A survey,, Nonlinearity, 23 (2010).  doi: 10.1088/0951-7715/23/4/R01.  Google Scholar [16] W. Cheng, Y. Zhao and Y. Cao, Pressures for asymptotically subadditive potentials under a mistake funciton,, Discret. Contin. Dyn. Syst., 32 (2012), 487.  doi: 10.3934/dcds.2012.32.487.  Google Scholar [17] V. Climenhaga, Bowen's equation in the non-uniform setting,, Ergod. Th. Dynam. Syst., 31 (2011), 1163.  doi: 10.1017/S0143385710000362.  Google Scholar [18] M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps,, Nonlinearity, 4 (1991), 103.  doi: 10.1088/0951-7715/4/1/008.  Google Scholar [19] K. Falconer, Fractal Geometry-Mathematical Foundations and Applications,, Second edition. John Wiley & Sons, (2003).  doi: 10.1002/0470013850.  Google Scholar [20] K. Falconer, Bounded distortion and dimension for non-conformal repellers,, Math. Proc. Camb. Phil. Soc., 115 (1994), 315.  doi: 10.1017/S030500410007211X.  Google Scholar [21] D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps,, Ergod. Th. Dynam. Syst., 17 (1997), 147.  doi: 10.1017/S0143385797060987.  Google Scholar [22] H. Hu, Dimensions of invariant sets of expanding maps,, Commun. Math. Phys., 176 (1996), 307.  doi: 10.1007/BF02099551.  Google Scholar [23] W. Huang and P. Zhang, Pointwise dimension, entropy and Lyapunov exponents for $C^1$ map,, Trans. Amer. Math. Soc., 364 (2012), 6355.  doi: 10.1090/S0002-9947-2012-05527-9.  Google Scholar [24] F. Ledrappier, Some relations between dimension and Lyapunov exponent,, Commun. Math. Phys., 81 (1981), 229.  doi: 10.1007/BF01208896.  Google Scholar [25] V. Mayer and M. Urbański, Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order,, Ergod. Th. Dynam. Syst., 28 (2008), 915.  doi: 10.1017/S0143385707000648.  Google Scholar [26] V. Mayer and M. Urbański, Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order,, Mem. Amer. Math. Soc., 203 (2010).  doi: 10.1090/S0065-9266-09-00577-8.  Google Scholar [27] V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Lyapnov exponents of dynamical systems,, Trudy Moskov. Mat. Obšž, 19 (1968), 179.   Google Scholar [28] Y. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications,, Chicago: University of Chicago Press, (1997).   Google Scholar [29] F. Przytycki, J. Letelier and S. Smirnov, Equivalence and topo- logical invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps,, Invent. Math., 151 (2003), 29.  doi: 10.1007/s00222-002-0243-x.  Google Scholar [30] F. Przytycki, J. Letelier and S. Smirnov, Equality of pressures for rational functions,, Ergod. Th. Dynam. Syst., 24 (2004), 891.  doi: 10.1017/S0143385703000385.  Google Scholar [31] F. Przytycki and M. Urbažski, Conformal Fractals: Ergodic Theory Methods,, London Mathematical Society, (2010).   Google Scholar [32] D. Ruelle, Repellers for real analytic maps,, Ergod. Th. Dynam. Syst., 2 (1982), 99.  doi: 10.1017/S0143385700009603.  Google Scholar [33] D. Ruelle, An inequality for the entropy of differential maps,, Bol. Soc. Bras. De Mat., 9 (1978), 83.  doi: 10.1007/BF02584795.  Google Scholar [34] H. Rugh, On the dimensions of conformal repellers. Randomness and parameter dependency,, Annals of Mathematics, 168 (2008), 695.  doi: 10.4007/annals.2008.168.695.  Google Scholar [35] M. Urbažski, On the Hausdorff dimension of a Julia set with a rationally indifferent periodic point,, Studia Math., 97 (1991), 167.   Google Scholar [36] M. Urbažski, Parabolic Cantor sets,, Fund. Math., 151 (1996), 241.   Google Scholar [37] M. Urbažski and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family,, Ergod. Th. Dynam. Syst., 24 (2004), 279.  doi: 10.1017/S0143385703000208.  Google Scholar [38] P. Walters, An Introduction to Ergodic Theory,, Springer-Verlag, (1982).   Google Scholar [39] C. Wolf, On the box dimension of an invariant set,, Nonlinearity, 14 (2001), 73.  doi: 10.1088/0951-7715/14/1/303.  Google Scholar [40] L. S. Young, Dimension, entropy and Lyapunov exponents,, Ergod. th. Dynam. Syst., 2 (1982), 109.  doi: 10.1017/S0143385700009615.  Google Scholar [41] Y. Zhang, Dynamical upper bounds for Hausdorff dimension of invariant sets,, Ergod. Th. Dynam. Syst., 17 (1997), 739.  doi: 10.1017/S0143385797085003.  Google Scholar [42] Y. Zhao and Y. Cao, Measure-theoretic pressure for subadditive potentials,, Nonlinear analysis, 70 (2009), 2237.  doi: 10.1016/j.na.2008.03.003.  Google Scholar [43] Y. Zhao, A note on the measure-theoretic pressure in subadditive case,, Chinese Annals of Math., 29 (2008), 325.   Google Scholar
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